Kate on "Too many MySQL connections and MySQL server has gone away"

We've been having a lot of issues with errors like "WordPress database error MySQL server has gone away for..." I included some examples below. We *think* this may be related to maxed out MySQL connections. We have a server of about 100+ WordPress sites running. Any ideas on how we can configure MySQL/the server to handle this better?

mysql> show status like '%onn%';
+--------------------------+---------+
| Variable_name            | Value   |
+--------------------------+---------+
| Aborted_connects         | 835     |
| Connections              | 3998523 |
| Max_used_connections     | 533     |
| Ssl_client_connects      | 0       |
| Ssl_connect_renegotiates | 0       |
| Ssl_finished_connects    | 0       |
| Threads_connected        | 3       |
+--------------------------+---------+
7 rows in set (0.00 sec)

[Wed May 29 13:21:44 2013] [error]  WordPress database error MySQL server has gone away for query SELECT ID\n\t\t\tFROM xy27abwp_posts\n\t\t\tWHERE guid = 'http://urlhere/wp-content/uploads/logo-MO2.png'\n\t\t\tLIMIT 1 made by require('wp-blog-header.php'), wp, WP->main, WP->send_headers, apply_filters('wp_headers'), call_user_func_array, UserAccessManager->redirect, UserAccessManager->getFile, UserAccessManager->_getFileSettingsByType, UserAccessManager->getPostIdByUrl, referer: http://urlhere.com/

[Wed May 29 13:21:44 2013] [error] WordPress database error MySQL server has gone away for query SELECT ID, post_name, post_parent, post_type FROM xy27abwp_posts WHERE post_name IN ('wp-content','uploads','logo-mo2-png') AND (post_type = 'page' OR post_type = 'attachment') made by require('wp-blog-header.php'), wp, WP->main, WP->query_posts, WP_Query->query, WP_Query->get_posts, WP_Query->parse_query, get_page_by_path, referer: http://urlhere.com/

[Wed May 29 13:25:44 2013] [error] WordPress database error MySQL server has gone away for query SELECT option_value FROM summer_options WHERE option_name = 'feedwordpress_diagnostics_show' LIMIT 1 made by require('wp-blog-header.php'), require_once('wp-includes/template-loader.php'), include('/themes/mytheme2010/courses.php'), get_footer, locate_template, load_template, require_once('/themes/mytheme2010/footer.php'), wp_footer, do_action('wp_footer'), call_user_func_array, debug_out_feedwordpress_footer, FeedWordPress::diagnostic_on, get_option, referer: http://urlhere.com/